Some Convergence Properties of Matrix Sets

A Switching Design Approach for Discrete-time Linear Hybrid Systems

“…However, if the system is switched attractive, the algorithm does terminate in a finite number of steps, although we do not have an upper bound of the step number. Other termination conditions may apply, for example, if all the singular values of matrices A 1 , · · · , A m are greater than or equal to one, then the system is by no means to be switched attractive [9], and we can terminate the algorithm in m steps (j ≤ m).…”

“…In particular, for switched linear systems where no state reset occurs and no discrete state transition restriction, the equivalence among (i), (ii), and (iii) has been addressed in [7] for continuous-time systems and in [9] for discrete-time systems. For the systems which can be made quadratically stable by means of proper switching signals, the existence of a quadratic Lyapunov function guarantees that sufficiently small structural perturbations do not affect the switched stability.…”

“…The switching strategy generates event-driven switching signals in state feedback form. Other approaches include the phase plane analysis that leads to the conic switching law for planar systems [3], [12], and the algebraic approach based on eigenvalue analysis [9], [4]. The reader is referred to [1], [5] for surveys of recent developments.…”

On stability of switched linear systems with perturbed switching paths

“…To reinforce the robustness performance and to reduce the switching frequency, a combined switching strategy was proposed for a class of switched linear systems [15]. Other existing approaches include the phase plane analysis that leads to the conic switching law for planar systems [16,17], and the algebraic approach based on eigenvalue analysis [18,19]. The reader is referred to [20∼22] for surveys of recent developments.…”

The set of stable switching sequences for discrete-time linear switched systems

“…A switched system is said to be asymptotically stable (also called absolutely stable) if its all admissible trajectories converge to the origin. The existing approaches in literature for showing absolute stability of switched systems, to our best knowledge, are essentially based on the search of common Lyapunov functions [3,4,10,12,18] or variations of the same framework [2,7,14,16,19,22], and the stability is characterized by the existence of positive definite solutions of a finite set of linear matrix inequalities (LMIs). Recent notable developments in this direction under the Lyapunov function framework include, among others, (i) to seek the largest set of switching sequences on which the system is asymptotically stable [13][14][15]17], and (ii) to determine the minimum dwell time such that a set of asymptotically stable switching sequences can be identified [5,6,21].…”